I must write a quadratic equation and i know only one zero. What is the quadratic whose zero is 2-2i?

We'll recall the fact that a quadratic equation has two
zeroes. We notice that the given zero is a complex number and we know that the complex
roots come in pairs. Therefore, we'll determine the other zero of the quadratic to be
found as being the conjugate of the given zero, namely 2 +
2i.

We'll write the factor form of a
quadratic:

(x - x1)(x - x2) = 0, where x1 and x2 represent
the zeroes of the quadratic.

We'll substitute x1 and x2 by
the given roots:

x1 = 2  -2i and x2 = 2 +
2i

(x - 2 + 2i)(x - 2 - 2i) =
0

We notice that the product above represents a special
product that returns a difference of two squares:

`(x - 2 +
2i)(x - 2 - 2i) = (x-2)^2 - (2i)^2`

We'll expand the
binomial:

`(x - 2 + 2i)(x - 2 - 2i) = x^2 - 4x + 4 -
4i^2`

But `i^2 = -1`

`(x - 2 +
2i)(x - 2 - 2i) = x^2 - 4x + 4 + 4`

`(x - 2 + 2i)(x - 2 -
2i) = x^2 - 4x + 8`

The requested quadratic,
whose roots are 2 - 2i and 2 + 2i, is `x^2 - 4x + 8 = 0.`